Internal dominance at fixed multiplicity and Frobenius number

Prove that, for integers m and F satisfying 2≤m≤F+1 and m∤F, the number of leaf numerical semigroups with multiplicity m and Frobenius number F is at most the number of internal numerical semigroups with the same multiplicity and Frobenius number.

Background

The final classification fixes multiplicity and Frobenius number simultaneously. Tables of computed values suggest that internal semigroups dominate leaf semigroups under the natural admissibility conditions on m and F.

References

Let $m$ and $F$ be two integer numbers such that $2\leq m\leq F+1$ and $m\not | F$, then $#\mathscr{L}(\text{mul}=m,\text{ Frob}=F)\leq #\mathscr{I}(\text{mul}=m,\text{ Frob}=F)$.

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Section 7, Internal numerical semigroups with fixed multiplicity and Frobenius number